If you want to win against archmage almost every time:
any fire cards (I suggest the three spells and fire elemental)
any water cards (I suggest meditation, water elemental, mind controller, astral guard)
any air cards (I suggest all four of the spells)
any earth cards, as long as you have master healer
choose illusionist: any illusion cards, as long as you have phantom warrior
now go into archduel against an archmage (these battles will count toward the achievements)
choose master healer as your special creature
just play phantom warriors (and some other good cards when available), with master healer, phantom warriors are nearly indestructible, always keep one master healer on the board, and you should win easily, use spells when needed.
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quick suggestion, change the location of the "hints" link, it covers some of the important pieces of the puzzle, for example, on lvl 13, the thing (i feel retarded, as i have no idea what it is) is covered by it, and it took quick a while to figure out what it was...